Journal of the Australian Mathematical Society - Series A
Vol. 64 Part 2 (1998)
Multiplicative isogeny estimates
The theory of isogeny estimates for Abelian varieties provides `additive bounds' of the form `d is at most B' for the degrees d of certain isogenies. We investigate whether these can be improved to `multiplicative bounds' of the form `d divides B'. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3).
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